2017 Convective Heat Transfer Final Exam  
Wednesday June 14th 2017
16:30 — 19:30


Question #1
Starting from the principle of conservation of mass, show that: $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0$$ with $\rho$ the mass density, and $u,v,w$ the $x,y,z$ components of the velocity vector.
Question #2
Consider two large (infinite) parallel plates, 5 mm apart. One plate is stationary, while the other plate is moving at a speed of $200$ m/s. Both plates are maintained at $27^\circ$C. Consider two cases, one for which the plates are separated by water and the other for which the plates are separated by air.
(a)  For each of the two fluids, what is the force per unit surface area required to maintain the above condition? What is the corresponding power requirement?
(b)  What is the viscous dissipation associated with each of the two fluids?
(c)  What is the maximum temperature in each of the two fluids?
Question #3
Consider a fin with rectangular cross-section attached to a wall maintained at a temperature $T_0$. The fin is cooled by a fluid with a convective heat transfer coefficient $h$ and a temperature $T_\infty$ (fluid temperature far from the fin). The fin has a length $L$, a depth $D$ and a thickness $t$. The cross-sectional area of the fin corresponds to $A=D\,t$:
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Given the thermal conductivity of the fin, $k$, and assuming that the convective heat transfer coefficient $h$ is constant over all the fin exposed surfaces, derive an expression for the conduction heat transfer at the base of the fin (i.e., where the fin is attached to the wall). Note: the fin tip is not insulated.
Question #4
You are walking on a frozen lake that has a radius of 50 m and you wish to determine the thickness of the ice layer without drilling through the ice. Remembering your heat transfer class, you rather decide to measure the temperature of the ice surface and the temperature of the air far from the ice. Knowing that there is no wind, that the air temperature far from the surface is of $-20^\circ$C, that the temperature of the ice touching the air is of $-10^\circ$C, find the thickness of the ice. You can use the following thermophysical data for ice, liquid water, and air:
PropertyIceLiquid waterAir
$c_p$, kJ/kgK24.21
$k$, W/mK2.30.60.02
$\mu$, kg/ms$10^{-3}$$10^{-5}$
$\rho$, kg/m$^3$92010001.2
Question #5
You wish to warm up cold water through an electrical heater. The electrical heater surrounds a water pipe and is insulated such that all of the heat generated by the heater is transfered to the water flowing in the pipe. Knowing that the pipe is made of copper and has an inner radius of 1 cm and an outer radius of 1.1 cm, that the electrical heater generates 200 W of heat per meter length of pipe, and that the water enters the pipe with a bulk velocity of 1 m/s and a bulk temperature of $30^\circ$C, determine the maximum length of the pipe that prevents the water from boiling; for a safe design the water should not approach the boiling point by less than $5^\circ$C at any location.
You can use the following thermophysical data for copper and liquid water:
PropertyCopperLiquid water
$c_p$, kJ/kgK0.44.2
$k$, W/mK3860.6
$\mu$, kg/ms$10^{-3}$
$\rho$, kg/m$^3$90001000
Question #6
Consider a scramjet combustor. The combustor is made of steel that is 1 cm thick. The hot gases exiting the combustor have an average temperature of 3000 K, a pressure of 2-atm, and an average velocity of 2000 m/s. As well, the boundary layer at the combustor exit is noticed to be turbulent and to have a height of 2 cm. You decide to cool the combustor wall at the combustor exit through film cooling. Film cooling consists of injecting liquid kerosene on all the inner surfaces of the combustor such that it evaporates when in contact with the hot gases and hence keeps the wall temperature to low values. For optimal design, it is here desired that the film cooling minimizes the amount of injected kerosene while keeping the inner combustor wall at a temperature not exceeding $200^\circ$C. Knowing that the latent heat of vaporization of kerosene is of $\Delta H_{\rm vap}=251~$kJ/kg and the saturation temperature of kerosene is of $T_{\rm sat}=200^\circ$C, and given the following properties for the gases and the steel:
Matter$\rho,$ kg/m$^3$$R,$ J/kg$^\circ$C$c_p,$ J/kg$^\circ$C$k,$ W/m$^\circ$C$\mu,$ kg/ms
Gases--33812000.1$6\cdot 10^{-5}$
Find the mass flow rate of injected kerosene per wall surface area at the combustor exit that results in an optimal design.
Hint: you can model the combustor wall physics as a 2D flow over a flat plate where the freestream properties correspond to the average properties at the combustor exit.
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